# Properties

 Label 2352.4.a.cn Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.391168.1 Defining polynomial: $$x^{4} - 40 x^{2} + 382$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: no (minimal twist has level 1176) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 2 + \beta_{1} ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 2 + \beta_{1} ) q^{5} + 9 q^{9} + ( -10 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( 12 + \beta_{2} + \beta_{3} ) q^{13} + ( -6 - 3 \beta_{1} ) q^{15} + ( 18 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{17} + ( -8 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{23} + ( 41 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{25} -27 q^{27} + ( 36 + 12 \beta_{1} + 5 \beta_{2} ) q^{29} + ( -12 - 6 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 30 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{33} + ( 12 + 14 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} ) q^{37} + ( -36 - 3 \beta_{2} - 3 \beta_{3} ) q^{39} + ( 18 - 3 \beta_{1} + 11 \beta_{2} + 20 \beta_{3} ) q^{41} + ( -128 - 2 \beta_{1} + 10 \beta_{2} + 10 \beta_{3} ) q^{43} + ( 18 + 9 \beta_{1} ) q^{45} + ( -40 - 2 \beta_{1} + 11 \beta_{2} - 6 \beta_{3} ) q^{47} + ( -54 + 9 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{51} + ( 134 + 12 \beta_{1} - 6 \beta_{2} + 24 \beta_{3} ) q^{53} + ( 84 - 6 \beta_{1} - \beta_{2} - 18 \beta_{3} ) q^{55} + ( 24 - 6 \beta_{1} - 15 \beta_{2} - 6 \beta_{3} ) q^{57} + ( -60 + 30 \beta_{1} - 7 \beta_{2} + 26 \beta_{3} ) q^{59} + ( 224 + 10 \beta_{1} + \beta_{2} - 29 \beta_{3} ) q^{61} + ( -34 + 14 \beta_{1} - 3 \beta_{2} - 22 \beta_{3} ) q^{65} + ( -272 + 22 \beta_{1} + 2 \beta_{2} - 38 \beta_{3} ) q^{67} + ( 6 + 9 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{69} + ( -322 + 13 \beta_{1} - 33 \beta_{2} + 25 \beta_{3} ) q^{71} + ( 372 + 18 \beta_{1} - 26 \beta_{2} + \beta_{3} ) q^{73} + ( -123 - 12 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{75} + ( -104 + 36 \beta_{1} - 20 \beta_{2} + 24 \beta_{3} ) q^{79} + 81 q^{81} + ( 28 + 32 \beta_{1} + 26 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -378 + 10 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} ) q^{85} + ( -108 - 36 \beta_{1} - 15 \beta_{2} ) q^{87} + ( 790 - 25 \beta_{1} + 39 \beta_{2} - 28 \beta_{3} ) q^{89} + ( 36 + 18 \beta_{1} + 27 \beta_{2} - 6 \beta_{3} ) q^{93} + ( 60 + 6 \beta_{1} + 10 \beta_{2} - 102 \beta_{3} ) q^{95} + ( 596 + 22 \beta_{1} - 24 \beta_{2} + 25 \beta_{3} ) q^{97} + ( -90 + 9 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10})$$ $$4q - 12q^{3} + 8q^{5} + 36q^{9} - 40q^{11} + 48q^{13} - 24q^{15} + 72q^{17} - 32q^{19} - 8q^{23} + 164q^{25} - 108q^{27} + 144q^{29} - 48q^{31} + 120q^{33} + 48q^{37} - 144q^{39} + 72q^{41} - 512q^{43} + 72q^{45} - 160q^{47} - 216q^{51} + 536q^{53} + 336q^{55} + 96q^{57} - 240q^{59} + 896q^{61} - 136q^{65} - 1088q^{67} + 24q^{69} - 1288q^{71} + 1488q^{73} - 492q^{75} - 416q^{79} + 324q^{81} + 112q^{83} - 1512q^{85} - 432q^{87} + 3160q^{89} + 144q^{93} + 240q^{95} + 2384q^{97} - 360q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 40 x^{2} + 382$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} - \nu^{2} + 40 \nu + 20$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{2} + 12 \nu + 40$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{2} - 140$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{3} + 7 \beta_{2}$$$$)/28$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{3} + 140$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$($$$$17 \beta_{3} + 70 \beta_{2} - 21 \beta_{1}$$$$)/14$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.92368 −3.96955 −4.92368 3.96955
0 −3.00000 0 −13.3405 0 0 0 9.00000 0
1.2 0 −3.00000 0 −7.81338 0 0 0 9.00000 0
1.3 0 −3.00000 0 14.5121 0 0 0 9.00000 0
1.4 0 −3.00000 0 14.6418 0 0 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.4.a.cn 4
4.b odd 2 1 1176.4.a.be yes 4
7.b odd 2 1 2352.4.a.co 4
28.d even 2 1 1176.4.a.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.4.a.z 4 28.d even 2 1
1176.4.a.be yes 4 4.b odd 2 1
2352.4.a.cn 4 1.a even 1 1 trivial
2352.4.a.co 4 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{4} - 8 T_{5}^{3} - 300 T_{5}^{2} + 1456 T_{5} + 22148$$ $$T_{11}^{4} + 40 T_{11}^{3} - 232 T_{11}^{2} - 4704 T_{11} + 7056$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T )^{4}$$
$5$ $$22148 + 1456 T - 300 T^{2} - 8 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$7056 - 4704 T - 232 T^{2} + 40 T^{3} + T^{4}$$
$13$ $$5508 + 8928 T + 124 T^{2} - 48 T^{3} + T^{4}$$
$17$ $$695428 + 44208 T - 1324 T^{2} - 72 T^{3} + T^{4}$$
$19$ $$50872896 - 213504 T - 14992 T^{2} + 32 T^{3} + T^{4}$$
$23$ $$-241264 - 104928 T - 3944 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$193813056 + 1033344 T - 44720 T^{2} - 144 T^{3} + T^{4}$$
$31$ $$280696384 + 2404992 T - 57904 T^{2} + 48 T^{3} + T^{4}$$
$37$ $$3279104 + 11171072 T - 95392 T^{2} - 48 T^{3} + T^{4}$$
$41$ $$-110754684 - 10848720 T - 143276 T^{2} - 72 T^{3} + T^{4}$$
$43$ $$-1048085504 - 16774144 T + 18368 T^{2} + 512 T^{3} + T^{4}$$
$47$ $$908321344 - 1189376 T - 88720 T^{2} + 160 T^{3} + T^{4}$$
$53$ $$520036624 + 4927648 T - 88104 T^{2} - 536 T^{3} + T^{4}$$
$59$ $$-10001985984 - 176574336 T - 448304 T^{2} + 240 T^{3} + T^{4}$$
$61$ $$-2069850492 + 50346624 T + 85436 T^{2} - 896 T^{3} + T^{4}$$
$67$ $$-23013868544 - 191398912 T - 46144 T^{2} + 1088 T^{3} + T^{4}$$
$71$ $$-188096048496 - 890221536 T - 419240 T^{2} + 1288 T^{3} + T^{4}$$
$73$ $$-21509659836 + 202647456 T + 197404 T^{2} - 1488 T^{3} + T^{4}$$
$79$ $$-88431606784 - 574626816 T - 862400 T^{2} + 416 T^{3} + T^{4}$$
$83$ $$1047073024 - 59481856 T - 608416 T^{2} - 112 T^{3} + T^{4}$$
$89$ $$-1254461582844 + 1366317456 T + 2135956 T^{2} - 3160 T^{3} + T^{4}$$
$97$ $$13301642052 - 244151136 T + 1344796 T^{2} - 2384 T^{3} + T^{4}$$